Erbil, Iraq—Brig. Halgord Hikmat, the spokesman for the Kurdistan Regional Government's Peshmerga Ministry, made little effort to conceal his satisfaction on Friday. "Yes," he said with a cheeky grin. "It is a very nice time to be Kurdish." Only a few hours before, Kurdish soldiers known as Peshmerga were engaging Islamist insurgent fighters in Diyala Province, picking up the slack from hundreds of retreating Iraqi troops. On Thursday Peshmerga fighters moved decisively to occupy the nearby city of Kirkuk, easily expelling Islamist fighters who had sent Iraqi troops literally running for the hills. The Iraqi government in Baghdad has suffered a string of humiliating military defeats this week as insurgents from the al Qaeda-inspired Islamic State of Iraq and al-Sham conquered several major cities. But for Iraq's long put-upon Kurdish minority, the past week has offered perhaps the finest opportunity in a generation to assert the Kurds' long-delayed claims on disputed lands and eventually, said Brig. Hikmat, a fully independent Kurdish state.